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Calc 3 Question on Curvature
My main question is on curvature and how to find the min/max. I haven't had Calc 1 in about 2 years, so some things are foggy to me at this point.
"At what point does y = ln(x) have maximum curvature? What happens to the curvature as x -> infinity"
You end up with the curvature equation:
k = -1/x<sup>2</sup> / (1 + 1/x<sup>2</sup>)<sup>3/2</sup>
To find the min max, you need to take the derivative of the curvature, which ends up being:
2/((1+1/x<sup>2</sup>)<sup>3/2</sup>x<sup>3</sup>) - 3/((1+1/x<sup>2</sup>)<sup>5/2</sup>x<sup>5</sup>)
...and this is where I'm stuck. How do I get the min/max from that equation? I remember something about setting that equal to 0 and testing points but... yeah, that's an algebraic mess to figure out.
I know based on limits that the curvature, as x approaches infinity, gets closer to 0 (1/x<sup>2</sup> gets closer to 0 as x gets bigger) .. at least I hope that part is right.
Based on that, I took a stab in the dark and said that the max curvature is at (1, 0). Somehow, without proof, I don't feel that's right.
Any help would be GREATLY appreciated!
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